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Magnetism in Matter

Magnetism in Matter. Electric polarisation ( P ) - electric dipole moment per unit vol. Magnetic ‘polarisation’ ( M ) - magnetic dipole moment per unit vol. M magnetisation Am -1 c.f. P polarisation Cm -2 Element magnetic dipole moment m

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Magnetism in Matter

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  1. Magnetism in Matter Electric polarisation (P) - electric dipole moment per unit vol. Magnetic ‘polarisation’ (M) - magnetic dipole moment per unit vol. Mmagnetisation Am-1 c.f. P polarisation Cm-2 Element magnetic dipole moment m When all moments have same magnitude & directionM=Nm N number density of magnetic moments Dielectric polarisation described in terms of surface (uniform) or volume (non-uniform) bound charge densities By analogy, expect description in terms of surface (uniform) or volume (non-uniform) magnetisation current densities

  2. Magnetism in Matter Electric polarisation P(r) Magnetisation M(r) p electric dipole moment of m magnetic dipole moment of localised charge distribution localised current distribution

  3. v4 r5 v5 v1 r4 O r1 r3 r2 v3 v2 Magnetic moment and angular momentum • Magnetic moment of a group of electrons m • Charge –e mass me

  4. Force and torque on magnetic moment

  5. B -e Diamagnetic susceptibility Induced magnetic dipole moment when B field applied Applied field causes small change in electron orbit, inducing L,m Consider force balance equation when B = 0 (mass) x (accel) = (electric force) wL is the Larmor frequency

  6. m a -e v -e v -e v x B -e v x B m Diamagnetic susceptibility Pair of electrons in a pz orbital B • = wo - wL |ℓ| = -mewLa2 m = -e/2me ℓ • = wo + wL |ℓ| = +mewLa2 m = -e/2me ℓ Electron pair acquires a net angular momentum/magnetic moment

  7. B -e m Diamagnetic susceptibility • Increase in angfreq increase in ang mom (ℓ) Increase in magnetic dipole moment: Include all Z electrons to get effective total induced magnetic dipole moment with sense opposite to that of B

  8. Paramagnetism Found in atoms, molecules with unpaired electron spins Examples O2, haemoglobin (Fe ion) Paramagnetic substances become weakly magnetised in an applied field Energy of magnetic moment in B field Um = -m.B Um = -9.27.10-24 J for a moment of 1 mB aligned in a field of 1 T Uthermal = kT = 4.14.10-21 J at 300K >> Um Um/kT=2.24.10-3 Boltzmann factors e-Um/kT for moment parallel/anti-parallel to B differ little at room temperature This implies little net magnetisation at room temperature

  9. Ferro, Ferri, Anti-ferromagnetism Found in solids with magnetic ions (with unpaired electron spins) Examples Fe, Fe3O4 (magnetite), La2CuO4 When interactions H = -J mi.mj between magnetic ions are (J) >= kT Thermal energy required to flip moment is Nm.B >> m.B N is number of ions in a cluster to be flipped and Um/kT > 1 Ferromagnet has J > 0 (moments align parallel) Anti-ferromagnet has J < 0 (moments align anti-parallel) Ferrimagnet has J < 0 but moments of different sizes giving net magnetisation

  10. I z y x Uniform magnetisation Electric polarisation Magnetisation Magnetisation is a current per unit length For uniform magnetisation, all current localised on surface of magnetised body (c.f. induced charge in uniform polarisation)

  11. m M Surface Magnetisation Current Density Symbol: aM a vector current density Units: Am-1 Consider a cylinder of radius r and uniform magnetisation M where M is parallel to cylinder axis Since M arises from individualm, (which in turn arise in current loops) draw these loops on the end face Current loops cancel in interior, leaving only net (macroscopic) surface current

  12. aM M Surface Magnetisation Current Density magnitude aM = M but for a vector must also determine its direction aM is perpendicular to both M and the surface normal Normally, current density is “current per unit area” in this case it is “current per unit length”, length along the Cylinder - analogous to current in a solenoid.

  13. B I L Surface Magnetisation Current Density Solenoid in vacuum With magnetic core (red), Ampere’s Law integration contour encloses two types of current, “conduction current” in the coils and “magnetisation current” on the surface of the core  > 1: aM and I in same direction (paramagnetic)  < 1: aM and I in opposite directions (diamagnetic)  is the relative permeability, c.f. e the relative permittivity Substitute for aM

  14. Magnetisation Macroscopic electric field EMac= EApplied + EDep = E - P/o Macroscopic magnetic field BMac= BApplied + BMagnetisation BMagnetisation is the contribution to BMac from the magnetisation BMac= BApplied + BMagnetisation = B + moM Define magnetic susceptibility via M= cBBMac/mo BMac= B + cBBMac EMac= E - P/o= E - EMac BMac(1-cB) = B EMac(1+c) = E Diamagnets BMagnetisation opposes BApplied cB < 0 Para, Ferromagnets BMagnetisation enhances BApplied cB > 0 B Au -3.6.10-5 0.99996 Quartz -6.2.10-5 0.99994 O2 STP +1.9.10-6 1.000002

  15. Magnetisation Rewrite BMac= B + moMas BMac - moM= B LHS contains only fields inside matter, RHS fields outside Magnetic field intensity, H = BMac/mo - M= B/mo = BMac/mo - cBBMac/mo = BMac(1- cB) /mo = BMac/mmo c.f. D = oEMac + P = oEMac m = 1/(1- cB)  = 1 + c Relative permeability Relative permittivity

  16. z I1-I2 I2-I3 My I1 I2 I3 x Non-uniform Magnetisation Rectangular slab of material with M directed along y-axis M increases in magnitude along x-axis Individual loop currents increase from left to right There is a net current along the z-axis Magnetisation current density

  17. dx dx Non-uniform Magnetisation Consider 3 identical element boxes, centres separated by dx If the circulating current on the central box is Then on the left and right boxes, respectively, it is

  18. Non-uniform Magnetisation Magnetisation current is the difference in neighbouring circulating currents, where the half takes care of the fact that each box is used twice! This simplifies to

  19. My -Mx I1-I2 I2-I3 z z y x I1 I2 I3 x Non-uniform Magnetisation Rectangular slab of material with M directed along x-axis M increases in magnitude along y-axis Total magnetisation current || z Similar analysis for x, y components yields

  20. k M= sin(ay) k j i jM= curl M = a cos(ay) i Types of Current j Total current Polarisation current density from oscillation of charges as electric dipoles Magnetisation current density from space/time variation of magnetic dipoles

  21. Magnetic Field Intensity H Recall Ampere’s Law Recognise two types of current, free and bound

  22. Magnetic Field Intensity H D/t is displacement current postulated by Maxwell (1862) to exist in the gap of a charging capacitor In vacuum D = eoE and displacement current exists throughout space

  23. 1 2 B1 q1 B2 q1 dℓ1 H1 1 B A C Ienclfree q2 2 q2 H2 dℓ2 S Boundary conditions on B, H For LIH magnetic media B = mmoH (diamagnets, paramagnets, not ferromagnets for which B = B(H))

  24. Boundary conditions on B, H

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